"To break the stalemate" means to end a situation where neither side is making progress or gaining an advantage. It involves taking action or finding a solution that changes the situation in order to move forward. Full definition
Having a thermodynamic route to the same universal end game could help break the stalemate over the mismatch problem. (newscientist.com)
1999 — NY State Attorney General Eliot Spitzer sues GE seeking monetary damages for PCB contamination stating that he wants to break the stalemate between GE and government regulators. (riverkeeper.org)
Despite being snakebitten in the first half, Shorewood was finally able to break the stalemate in the second half thanks to goals by Hayk Avanesayan, Eugene Holley and Daniel Yun. (maverickfootballclub.com)